When this research was performed, P. Fritschel, N. Mavalvala, D. Shoemaker, D. Sigg, and M. Zucker were with the Department of Physics and Center for Space Research, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, and

G. González was with the Department of Physics, Pennsylvania State University, University Park, Pennsylvania 16802.

N. Mavalvala is now with the LIGO Project, California Institute of Technology, MS 18-34, Pasadena, California 91125, and

Abstract

Interferometric gravitational wave detectors are designed to detect
small perturbations in the relative lengths of their kilometer-scale
arms that are induced by passing gravitational radiation. An
analysis of the effects of imperfect optical alignment on the strain
sensitivity of such an interferometer shows that to achieve maximum
strain sensitivity at the Laser Interferometer Gravitational Wave
Observatory requires that the angular orientations of the optics be
within 10-8 rad rms of the optical axis, and the beam must
be kept centered on the mirrors within 1 mm. In addition,
fluctuations in the input laser beam direction must be less than
1.5 × 10-14 rad/Hz in angle and less
than 2.8 × 10-10 m/Hz in transverse
displacement for frequencies f > 150 Hz in order that
they not produce spurious noise in the gravitational wave readout
channel. We show that seismic disturbances limit the use of local
reference frames for angular alignment at a level approximately an
order of magnitude worse than required. A wave-front sensing scheme
that uses the input laser beam as the reference axis is presented that
successfully discriminates among all angular degrees of freedom and
permits the implementation of a closed-loop servo control to suppress
the environmentally driven angular fluctuations
sufficiently.

References

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a For each angular degree of freedom, the
top number listed is Ai (in watts, with
significant values in boldface); below left is the RF phase
ϕDi, in phase (I) or in quadrature (Q)
with the modulation phase at the input, and below right is the Gouy
phase angle ηi. The last row lists
the signals produced with phase-modulation sidebands that are not
resonant (NR sideband) in the interferometer.

Table 3

The Five Wave-Front Sensor Signals for a Specific Choice
of Sensor Positions, Gouy Phases, and RF
Phasesa

a This set of five signals gives a
nondegenerate sensing matrix for all the relevant angular degrees of
freedom. The magnitude of these signals,
AWFS, is given in watts and is normalized in the
same way as the Ai in Table 2.
b
I, in phase; Q, in quadrature.

a The splitting factors for the sensors at
the antisymmetric and the reflected ports (wave-front sensors 1, 3,
and 4) are small, so most of the light can be directed to the length
sensors; for the recycling cavity port sensor (wave-front sensors 2a
and 2b), the splitting factor is unity because there are other
antireflection surface beams available for the length
sensor. Sensors 3 and 4 use the nonresonant sideband for detection,
for which the modulation index is chosen to be 1/10 that of the main
modulation frequency.

Tables (4)

Table 1

Eigenvalues and Eigenvectors of the Five-Dimensional
Misalignment Variance Ellipsoid for the Shot-Noise-Limited
Signal-to-Noise Ratio of the
Interferometera

i

Eigenvalue σi
2

Eigenvector (Ellipsoid Axis)

ΔETM

ΔITM

ETM¯

ITM¯

RM

1

0.00061

0

0

0

-0.58

0.81

2

0.00050

0.91

0.42

0

0

0

3

0.12

-0.42

0.91

0

0

0

4

0.83

0

0

0.92

0.32

0.23

5

6.4

0

0

-0.39

0.75

0.54

a The eigenvalues are in units of (the
square of) the beam divergence angle. Significant values are in
boldface.

a For each angular degree of freedom, the
top number listed is Ai (in watts, with
significant values in boldface); below left is the RF phase
ϕDi, in phase (I) or in quadrature (Q)
with the modulation phase at the input, and below right is the Gouy
phase angle ηi. The last row lists
the signals produced with phase-modulation sidebands that are not
resonant (NR sideband) in the interferometer.

Table 3

The Five Wave-Front Sensor Signals for a Specific Choice
of Sensor Positions, Gouy Phases, and RF
Phasesa

a This set of five signals gives a
nondegenerate sensing matrix for all the relevant angular degrees of
freedom. The magnitude of these signals,
AWFS, is given in watts and is normalized in the
same way as the Ai in Table 2.
b
I, in phase; Q, in quadrature.

a The splitting factors for the sensors at
the antisymmetric and the reflected ports (wave-front sensors 1, 3,
and 4) are small, so most of the light can be directed to the length
sensors; for the recycling cavity port sensor (wave-front sensors 2a
and 2b), the splitting factor is unity because there are other
antireflection surface beams available for the length
sensor. Sensors 3 and 4 use the nonresonant sideband for detection,
for which the modulation index is chosen to be 1/10 that of the main
modulation frequency.